Injury, Recovery and Death in Relation to Conductivity and Permeability

v It is recognized that the hypothesis will apply if the layer is not continuous and also if the change in properties of the layer is other than that of thickness. 30 These reactions are regarded as reversible or practically so. curve shown in the figure, while if K1 is less than K2 we get the lower curve (in the latter case both outlets are supposed to be smaller than in the former). This is analogous to what occurs in the reaction A — >M — >B if Kl is the velocity constant of A — >M and K2 is the velocity constant of M — >B.

FIG. 27. — Diagram illustrating consecutive reactions in which a oubsf anoe M in formed by the reaction A — >• M and decomposed by the reaction M— >•£. as it is decomposed and has the constant value 2700, and that M has the constant value 90. On transferring from sea water to NaCl 0.52 M the production of A ceases, but A continues to break down to form M and B. The velocity constants31 in NaCl are taken as Kl=O.Q'L8 and 7ir2=0.540. We may now calculate the resistance, putting Net Resistance=M+10, 'because the base line of the death

curve, as shown in Fig. 28, is not zero but ten. M is therefore equal to the resistance of the living tissue after the resistance of the dead tissue and the resistance of the apparatus has been subtracted. We may call this the residual resistance, while the resistance of the living tissue minus that of the apparatus is called the net resistance. Fio. 28. — Death curve of Laminaria agardhii in NaCl 0.52 M. The curve shows the cal- culated value of the resistance: the observed values are shown by the points (O. O)- All readings were made at 15° C. or corrected to this temperature. Each point represents th« average of ten or more experiments. Probable error of the mean less than 10% of the mean.

In our calculations we may employ the methods used in calculating the decomposition of radioactive substances.32 If we start with A and M in equilibrium in sea water according to the scheme A — >M — >B, and if the value of A is called A0 and that of M is called M0, the amount of M which is formed in each unit of time is A0K^ and the amount of M which decomposes in each unit of time is M0K2. Since at equilibrium, AQK^=M^K^ the value of M remains constant.

the instant of transfer the following values : A0 = 2700, M0=90, ^=0.018, and K2=OMQ. As no more of A is produced A0 diminishes and at the end of time T has the tain amount of M is formed from A. Some of this dis- appears during exposure. The amount which remains at the time T may be called M r At the time T the amount of MT which disappears in unit time is MTK2. At this time the amount of M produced in unit time is The change in M T occurring in unit time, which we may call ~~f is equal to the difference between the amount

We must also consider that at the beginning of the exposure to NaCl 0.52 M there was present a certain amount of M (this was called M0) which diminished dur- ing exposure, and the amount remaining at the time T is M0e 2 . If this is added to the amount of M pro- duced from A during exposure we get (substituting the value M0=90) and since Net Resistance - - M + 10 (because the base line of the curve is taken as 10) we have 33 If we calculate the resistance by means of this for- mula we get the curve given in Fig. 28, which shows a close agreement between the observed and calculated values. It is therefore evident that, whether our picture of the underlying mechanism is correct or not, it leads to an equation which enables us to predict the death curve with considerable accuracy. The predictive value of the equation is quite independent of the assumptions which led up to it, and while it creates a presumption in favor of these assumptions, it of course does nothing more. It is hardly necessary to emphasize that equations which enable us to predict the course of vital pro- cesses are a prime necessity in biology, since they make it possible to employ the methods by which the exact sciences have been able to make rapid progress.

in the Smithsonian Mathematical Tables, Hyperbolic Functions, by G. F. Becker and C. E. Van Orstrand, 1909. See also Mellor, J. W. (1909) pp. death in this case just as a chemist follows the progress of a reaction in vitro. This has also been found to be the case in experiments with a great number of toxic sub- stances and seems to be of very general applicability. Net electrical resistance of Laminaria in NaCl 0.52 M. The resistance in sea water (the normal environment) is taken as 100 per cent.

* All readings were made at 15°C., or corrected to this temperature. If we were unaware that the death curve in NaCl 0.52 M represented two consecutive reactions, and supposed it to represent a simple monomolecular reaction (M — > jB), we should calculate its velocity constant (which we may call K^) by the usual formula:34 M Common logarithms are used for convenience, We put a = 100 — 10 Mid » — x = M — 10, pose the calculated values given in the third column of Table I, we obtain the values of the velocity constant K3 given in the fifth column of the table.

It is evident from an inspection of these values that the velocity constant K3 falls below the average value at the start. The amount by which it falls below the average value will depend on the relation Kl -r- K2. When K± and K2 are nearly equal, the velocity constant falls a good deal below the average value at the start, but as the difference between them is increased the velocity constant K3 will be found to fall less and less below the average level at the start.35 This is easily shown by assuming various values36 of Kl and K2.

From this it follows that we can tell something about K1 -f- K2 from the experimental values of K3. It is evident that in the present case the experimental values of K point to the relation K2 -=- Kl = 30 (or Kl-±-K2 = 30). This relation was actually assumed by the writer, in order to fit, not the NaCl curve, but antagonism curves37 in various mixtures of NaCl + CaCl2. It is therefore a striking confirmation of the general correctness of the underlying assumption that we are also able by means of this assumption to fit the NaCl curve so closely.

In general, where a chemical reaction is slower at the start than is expected, we may suspect that we have " It should be noted that we get the same result (as .regards K, falling below the average at the start) when .K^ -^ ITj = 30 as when JBT, ~ Kt = 30. With certain relations of K^ -+- K2 the constant K3 may be above the average value at the start. M When the values of K^ and K3 are changed, the concentrations of A and M must also be changed in such a way that Cone. A •+- Cone. M = K! -f- Kt if we wish the concentrations of A and M to remain constant in the normal environment.

to do, not with a simple reaction, but with consecutive reactions of the land here described.38 This explanation also applies to a considerable num- ber of other cases of toxic action. FIQ. 29. — Curve showing the net electrical resistance of Laminaria agardhii in CaCh 0.278 M. Unbroken line, observed values; broken line, calculated values. All observations were made at 18° C. or corrected to this figure. Average of ten or more observations. Probable error

haves as a reaction which is continually going on, but at a very slow rate until accelerated by the toxic agent. We have assumed this acceleration to consist partly in the increase of the velocity constant and partly in the stopping of the reaction 0 — >A, causing a decrease in the sub- stance (M) to which normal permeability (and perhaps other normal properties) are due. It may prove to be generally true that death behaves as a monomolecular reaction, which is inhibited (or accelerated) at the start. The assumption of consecutive reactions affords an explanation not only of the inhibi- tion (or acceleration) at the start, but also of the fact that up to a certain point the reaction appears to be

Net electrical resistance of Laminaria in CaCl2 0.278M. The resistance in sea water (the normal environment} is taken as 100%. * The measurements were made at 15° Centigrade or corrected to this figure. Each experimental figure is the average obtained from 6 or more experiments. Probable error of the mean less than 10 % of the mean. reversible. The latter fact will be fully discussed in a subsequent chapter. It is evident that if the theory of the writer is sound, the equation which allows us to predict the death curve in experiments with NaCl should apply equally well in the case of experiments with CaCl2. This is the case, as is evident from Fig. 29 and Table II (in this case we put

maximum may be approximately ascertained by means of the formula.39 The actual maximum found by calculating the curve is close to 153.94 (which occurs at 75 minutes). Such a close approximation must not, however, be expected in most cases. Where the maximum of the curve is known and it is desired to find the relation Kl-~K.l (as a preliminary step toward ascertaining the values of Kl and K% by trial) we may plot a series of values of K^-~-K^ as ordinates, and maxima (obtained by calculation) as abscissae, and thus approximate graphically to the desired figure.

When the height of the maximum is known, the time at which the maximum occurs may be found as follows : When the maximum is attained the value of M may be called M max and the value of A may be called AT[ 39 This may be regarded as an approximation formula. We consider that the value of A before any of it has decomposed to form M and B, is 3050 and if this is substituted for 2830 in the formula it will give exact values, provided the constants are not changed as M increases from 0 to the maximum. But if M increases from 0 to 90 with one set of constants and then from 90 to the maximum with another set, the formula no longer holds and the approximation formula may be used. Cf. Mellor, J. W. (1909) p. 115. In the formula as given by Mellor a misprint occurs.

Since at this time M is formed as rapidly as it is decom- posed, we have At the start (in sea water) the value of A was 2700, but it has now diminished to a fraction represented by it requires to reach this value, for, as the reaction A — > M is monomolecular, we may write *°E.g., Table IV in the Smithsonian Mathematical Tables, Hyperbolic Functions, by G. F. Becker and C. E. Van Orstrand, 1909. See also Van Orstrand, C. E. (1921). same factor is equivalent to dividing all the abscissae by the same factor and that in the case of a curve which rises and falls, this does not change the height of the maximum. If, therefore, both reactions have the same temperature coefficient, raising the temperature is equiv- alent to multiplying both Kl and K2 by the same factor and the maximum will not be changed. But if the reac- tions have different temperature coefficients this will not be true.

Perhaps it may be desirable in this connection to add a word regarding the measurement of life processes. The development of quantitative methods in biology depends largely on finding means of measuring the speed of life processes. In most cases the absolute rate is of less importance that the relative rate (e. g., the normal veloc- ity compared with that observed under the influence of a reagent). Examination of the literature shows that the determination of relative rates is frequently made in a faulty manner, which might easily be avoided by a slight change of method.

As an illustration of this we may consider the processes shown in Fig. 30. In the case of Curve A the process is twice as rapid as in the case of Curve B. This is shown by the fact that the abscissae of A are everywhere one-half those of B. This means that the velocity constants of A are twice those of B.41 In other words the velocity con- stants are inversely proportional to the abscissae, or in- versely proportional to the times required to bring the reaction to the same stage42 (e.g. one-half com- pleted). This is true for chemical processes in general, not only for reactions of the first order (where a

single substance decomposes) but for reactions of higher orders (where two or more substances combine) as well as for consecutive reactions43 and autocatalysis-44 It follows that when a chemical process proceeds at different rates under different conditions, we can com- pare the velocity constants by simply taking the recip- rocals of the times required to bring the reaction to the same stage, so that if we wish to know merely the relative FIQ. 30. — Curve A represents a process which proceeds at twice the velocity of B. The abscisses of B are everywhere double those of A, but no such relation holds for the ordinates. C is obtained by averaging the abscissse of A and B; D is obtained by averaging their ordinates.

rates (as is usually the case in biology) it is not necessary to determine the actual velocity constants at all. Whenever the initial conditions are the same with respect to concentration we need only compare the times required for equal amounts of work, since these bring the reaction to the same stage. If, on the other hand, one attempts to arrive at the relative rate by comparing the amounts of work performed in equal times (as is fre- quently done in biological research) he can easily fall

48 The principle holds for consecutive reactions in case all the constants are multiplied by the same factor, otherwise not. Cf. Osterhout (1917, E). 44 Cf. Mellor (1909) p. 291. into serious error. This is evident from Fig. 30 which shows that while the abscissa of A at any point is just half that of B, no such relation obtains among the ordin- ates.46 For example at 40 minutes, the ordinate of B is twice as great as that of A, while at 4 minutes, it is less than 1.1 times that of A. Hence it is evident that we should compare abscissae rather than ordinates (i.e., times required to do equal amounts of work rather than amounts of work performed in equal times).

The principle is sufficiently obvious where successive determinations are made and curves are drawn. But there is a common type of experimentation in which, for various reasons, a single observation at one rate is com- pared with a single observation at another rate. The principle in question is then easily overlooked. In some cases this leads to serious errors.48 It is therefore evident that when we average time curves, we should, whenever possible, average abscissae rather than ordinates. Thus for example, in Fig. 30 the average of Curves A and B would be Curve C, obtained by averaging the abscissae of Curves A and B : this gives a curve whose velocity constants are the arithmetical mean of those of A and B. On the other hand, by averaging ordinates we obtain Curve D, which does not follow the formula characteristic of the other two curves.

It may be desirable to point out that these methods may be advantageously applied to the measurement of toxicity.47 46 We cannot avoid the difficulty by comparing the rates of the two processes at a given time; for the rates so obtained will bear no constant ratio to each other. Only when they are compared at the same stage of the reaction will they show a constant relation; this gives the relation between the velocity constants. One striking result of the investigations on toxicity carried out by the writer is to emphasize the fact that the apparent toxicity of two substances may depend very largely upon the stage of the reaction at which the meas-

Fio. 31. — Curves showing changes in the net electrical resistance of tissues in two toxic solutions, A and B (the latter causes a rise followed by a fall in resistance). Toxicity may be measured by determining the time required to carry the reaction to a definite stage, as, for example, to 55% which is half way between the normal condition and the death point. urement is made. This is evident from an inspection of the curves in Fig. 31. These represent the electrical resistance of Laminaria in sea water and in two toxic solutions. If the tissue be placed in a solution of NaCl of the same conductivity as sea water, the resistance falls, somewhat as shown in Curve A, until it reaches the death

point. If, on the other hand, the tissue be placed in a solution of some substance which causes a rise, followed by a fall in resistance, we may get a curve somewhat like that shown at B. The most common method of measuring the toxicity of a solution is to determine the time necessary to cause death. But it is evident from an inspection of the curves that it is impossible to determine the precise moment of death, since the death curves approach the axis asymptot- ically. This is doubtless true of death in all cases. It

is therefore obvious that the death point does not offer a perfectly satisfactory criterion of toxicity. We may avoid this difficulty by taking as a criterion the time needed to reach any convenient point on the curve, as, for example, 55% (half way between the normal condition and the death point). This may be determined with a good deal of precision by the measurement of elec- trical resistance or by any method which permits us to follow the reaction accurately from moment to moment. But where this cannot be done, we may employ other criteria. We may assume that as the reaction goes on, certain phenomena appear at definite points on the curve, such, for example, as changes in metabolism, cessation of motion, or loss of irritability. The employment of such criteria may give trustworthy results in many cases if proper precautions be taken.

In the employment of any of these criteria, except that of death, we may meet the difficulty that the relative toxicity of two substances may vary greatly according to the point in the curve at which the comparison is made. Let us suppose that two toxic substances are so chosen that they produce death at about the same time, giving curves as shown in Fig. 31. They must be regarded as equally toxic if we adopt death as the criterion, but as unequally toxic if we take any other criterion. For example at 90%, A appears to be seven times as toxic as B.

It is clear that we cannot escape from this difficulty by comparing the effects produced in equal times. In view of these facts it is obviously undesirable to compare results obtained by the use of unlike criteria, as is often done. Another method is to measure the degree of recovery which is found where tissues are taken from toxic solu- tions and replaced in sea water. This will be explained more fully in Chapter III. It has many advantages which entitle it to serious consideration. The writer has found that death in many toxic substances, as measured by the electrical method, follows approximately the course of a monomolecular reaction. In such cases the constants which express the reaction velocities of the two reactions afford a measure of their relative toxicity. In cases where such constants cannot be used, but where the com- plete curve can be obtained, it would be possible to adopt, as an arbitrary standard, the time necessary for the reaction to proceed half way to the death point. But, when the curves are related to each other as are A and B in Fig. 31, it may be desirable to use some other criterion. It is in any case desirable to give the whole curve, when- ever possible, so that the reader may apply his own criterion. The ease with which complete curves can be obtained by determining electrical resistance may render this method useful, especially since the writer has found it possible to apply it to all sorts of plant tissues as well as to some animal tissues.

The electrical method is not restricted to solutions of the same conductivity. For example, we find that NaCl 0.52 M and CaCl2 0.278 M have the same conduct- tivity as sea water. If we wish to compare the toxicity of NaCl 0.278 M with that of CaCl2 0.278 M we may dilute the sea water until it has the conductivity of NaCl 0.278 M. Tissue placed in this may be used as a control. At the outset we make the resistance of the control equal to that of the tissue in NaCl 0.278 M, or we divide the resistance of the control by a figure which reduces it to the same value (and divide all subsequent readings of the control by the same figure). We then express all readings of the tissue in NaCl 0.278 M as per cent, of the reading of the control which is taken at the same time. All readings of the tissue in CaCl2 0.278 M are likewise expressed as percentage of the readings of a control in sea water having the same conductivity as CaCl2 0.278 M. Stronger solutions may be treated in the same way, using sea water which has been concentrated by evaporation.